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This article was downloaded by: [Pennsylvania State University]On: 07 June 2013, At: 13:17Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
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An efficient domain decompositionframework for accurate representationof geodata in distributed hydrologicmodelsMukesh Kumar a , Gopal Bhatt a & Christopher J. Duffy aa Department of Civil and Environmental Engineering, Penn StateUniversity, University Park 16802, PA, USAPublished online: 04 Dec 2009.
To cite this article: Mukesh Kumar , Gopal Bhatt & Christopher J. Duffy (2009): An efficient domaindecomposition framework for accurate representation of geodata in distributed hydrologic models,International Journal of Geographical Information Science, 23:12, 1569-1596
To link to this article: http://dx.doi.org/10.1080/13658810802344143
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Research Article
An efficient domain decomposition framework for accuraterepresentation of geodata in distributed hydrologic models
MUKESH KUMAR*, GOPAL BHATT and CHRISTOPHER J. DUFFY
Department of Civil and Environmental Engineering, Penn State University, University
Park 16802, PA, USA
(Received 22 March 2008; in final form 11 June 2008 )
Physically-based, fully-distributed hydrologic models simulate hydrologic state
variables in space and time while using information regarding heterogeneity in
climate, land use, topography and hydrogeology. Since fine spatio-temporal
resolution and increased process dimension will have large data requirements,
there is a practical need to strike a balance between descriptive detail and
computational load for a particular model application. In this paper, we present
a flexible domain decomposition strategy for efficient and accurate integration of
the physiographic, climatic and hydrographic watershed features. The approach
takes advantage of different GIS feature types while generating high-quality
unstructured grids with user-specified geometrical and physical constraints. The
framework is able to anchor the efficient capture of spatially distributed and
temporally varying hydrologic interactions and also ingest the physical
prototypes effectively and accurately from a geodatabase. The proposed
decomposition framework is a critical step in implementing high quality,
multiscale, multiresolution, temporally adaptive and nested grids with least
computational burden. We also discuss the algorithms for generating the
framework using existing GIS feature objects. The framework is successfully
being used in a finite volume based integrated hydrologic model. The framework
is generic and can be used in other finite element/volume based hydrologic
models.
Keywords: Domain decomposition; Mesh generation; Constrained delaunay
triangulation; Hydrologic modeling; Geodata representation
1. Introduction
Distributed models simulate hydrologic states in space and time while using
discretized information regarding the distribution and parameters of climate, land
use, topography and hydrogeology (Freeze and Harlan 1969). These models have
inherent advantages over conventional lumped models particularly because natural
heterogeneities control watershed behavior(s) and also help in resolving the
feedback processes between state variables (Entekhabi and Eagleson 1989, Pitman
et al. 1990). The numerical solution strategies require spatial discretization of the
model domain into spatially connected units. For example, grid decomposition for
land surface models may take advantage of relevant physical subdomains such as
hillslopes (Band 1986), a contour (Moore et al. 1988), structured (Panday and
*Corresponding author. Email: [email protected]
International Journal of Geographical Information Science
Vol. 23, No. 12, December 2009, 1569–1596
International Journal of Geographical Information ScienceISSN 1365-8816 print/ISSN 1362-3087 online # 2009 Taylor & Francis
http://www.tandf.co.uk/journalsDOI: 10.1080/13658810802344143
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Huyakorn 2004) or unstructured grids (Qu and Duffy 2007). In the case of
multiprocess/multiscale models, the representation of topography, land cover, soil,
geology, vegetation and climate on a distributed model grid must, by necessity, deal
with questions of computational efficiency and limits of parameterization. Since our
goal is to perform physics based simulations on large watersheds, our strategy is to
minimize the resolution of spatial discretization (the fewest number of elements to
preserve the essential physics) while still capturing the local heterogeneities in
parameters and process dynamics. We achieve this by generating conformed Delaunay
triangulation using distributed GIS anchor objects like points, lines and polygons.
2. Domain decomposition: limitations and scope
Geometrically, the quality (shape and size) and type of the discrete elements which
make up the model grid determine the accuracy, convergence, memory storage and
computational cost of the numerical solution. Physically, the decomposed domain
must admit conformity with the boundary, connectivity between elements and
continuity of mass within each terrain element. The use of rectangular grids with
uniform topologic structure have seen wide use for domain discretization for
integrated hydrologic models such as PARFLOW-Surface Flow (Kollet and Maxwell
2006) and MODHMS (Panday and Huyakorn 2004). The inherent simplicity of a
structured grid has the advantage of fast computations for linear/nonlinear systems
because of uniformity in the size of neighboring grids and the ease of determining
grid’s neighbors. Furthermore, the regularity of structured meshes makes it
straightforward to parallelize computations. More to the point, it also complements
the data ingestion process for spatially distributed geologic, topographic and
meteorologic data that are available as raster maps/images. The computational
advantage of modeling on structured grids is sometimes offset by the need for very fine
spatial discretization in order to capture local heterogeneities and boundary ‘edges’.
Rigidness of the regular grids prevents the resolution of relatively small topological
structures without either resorting to a higher spatial resolution or using a nested or
adaptive mesh refinement (Blayo and Debreu 1999). Structured meshes also lack
flexibility in fitting complex-shaped domains. Techniques have been devised to find
appropriate coordinate transformations like conformal mapping and algebraic
methods which would lead to better fitting of irregular shapes (Castillo 1991,
Thompson 1982). However, these methods are complex and introduce errors due to
interpolation of derivatives. The ‘Staircase effect’ at the boundaries is observed when
no transformation is applied. No matter how fine the resolution of the regular grid is,
linear features which are not aligned in the principal direction of the grid are ‘aliased’.
This redefinition of boundaries because of gridding at different spatial scales,
particularly at places where hydrologic properties undergo a transition like a land
cover change from forest to urban or a topographic changeover, significantly affects
the modeling of movement of matter and energy (Woo 2004). Structured mesh
representations also restrict surface flow directions to 45u increments (Tarboton 1997)
which can introduce anisotropy in preferred flow direction (Braun and Sambridge
1997). Finally, regular grids create complications for Neumann-type boundary
conditions as they are forced to align in the two principal orthogonal directions along
which the grid is oriented. Figure 1 provides an example of the aliased boundary
representation for the Little Juniata watershed in central Pennsylvania.
Many of these limitations can be overcome using a decomposition strategy based
on Triangular Irregular Networks (TINs) (Goodrich 1990, Jones et al. 1990, Vivoni
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et al. 2004, Ivanov et al. 2004) or generalized unstructured meshes (Qu and Duffy
2007). The advantage of an unstructured mesh is that it can provide an ‘optimal’
representation of the domain with the least number of elements while still
conforming to a limited set of physical and geometric constraints. The unstructured
grid leads to a large decrease in the number of nodes/elements with respect to
structured meshes (Shewchuk 1996). Also, it allows better representation of line-
features such as the stream network, land-use/ land-cover boundaries and watershed
boundaries. In order to unlock the full potential of unstructured mesh decomposi-
tion, we generate them in a ‘smart’ way using GIS feature objects. This approach
leads to discrete domains that are computationally efficient and which honor the
edges and transitions of the important physiographic, geologic, ecologic and
hydroclimatic variables. The next section discusses unstructured grid representations
for integrated hydrologic modeling.
3. Solving multiple processes on an unstructured mesh
There are two approaches to solving systems of hydrologic equations simulta-
neously. The first is a weakly coupled approach where water and energy exchange
between surface water, groundwater, vegetation and atmosphere are solved
separately on different discretized domains while sharing interaction flux as
individual boundary conditions. The physical interaction in this case is weakly
coupled in that the communication between processes is intermittent and only occurs
as necessary to satisfy conservation or efficiency constraints. The synchronization of
communication is performed by controller software. Since the decomposition frame-
work for each individual physical process is separate, data assignment, topology
definitions, data geo-registration and flux exchange between different physical
components of the models is an error prone and computationally intensive
Figure 1. Domain decomposition of Little Juniata Watershed using rectangular grid. The‘zoomed-in’ portion of the boundary shows aliasing (staircase effect) of watershed boundaryby structured mesh. Such aliasing can be expected near all kinds of topographic, hydrographic(river, lakes, etc.) and physiographic (land use/land cover, soil types, etc.) boundaries.
Domain decompostion for geodata representation 1571
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approach. Weakly coupled models are also susceptible to convergence problems(Abbot et al. 1986) and unreliable solutions (Fairbanks et al. 2001). The second
strategy can be referred to as ‘full’ volume coupling, where all the physical process
equations are solved simultaneously on each element distributed across the
domain. For the purposes of this paper, there are important advantages to the
volume-coupling approach in that the approach offers a consistent and uniform
assignment and registration of geodata for all the physical model equations and
for all discrete elements in the domain. The Penn State Integrated Hydrologic
Model (PIHM; Kumar 2009, Qu and Duffy 2007) uses the latter approach and isbriefly described next.
PIHM is a semi-discrete finite volume method (FVM) based numerical model. It
solves ordinary differential equations (ODE) corresponding to all the interacting
hydrologic processes on each discretized watershed element. PIHM uses implicit
Newton–Krylov integrator available in the CVODE (Cohen and Hindmarsh 1994)
to solve for state variables at each time step. The discretized control volume
elements used in PIHM are either triangular or linear in shape as shown in Figure 2.
Triangular land surface elements are projected downwards to the bedrock or
Figure 2. All the interacting hydrologic processes in PIHM are defined on prismaticwatershed elements or linear river elements. ODEs corresponding to each process from allacross the model domain are solved together to predict state variables at next time step.
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regolith to form a prismatic element in three-dimension. Linear elements represent
rivers and are projected downwards to the river bed. The model is designed to
capture ‘dynamics’ in multiple processes while maintaining the conservation of mass
at all cells, as guaranteed by the finite volume formulation. The finite volume
formulation also has the ability to handle discontinuous solutions (Leveque 2002).
The conservation laws that are conveniently derived from the physical relationships
approximate the average state of the process variable over the kernel volume.
In order to perform accurate and efficient simulations, domain decomposition
quality should be considered as important as the numerical scheme itself. Mesh size,
shape and the ability to capture graded and/or sharp spatial changes depending on
the time scale of the interacting hydrologic processes, spatial gradient of the state
variables and the heterogeneity in model parameter distribution determine the
stability, convergence and accuracy of the solution. PIHM uses a smart unstructured
mesh decomposition that is generated using GIS feature objects to enhance
topographic and hydrographic representations, which are discussed in the next
section.
4. Unstructured mesh generation and GIS objects
Although theoretical and computational aspects of unstructured meshes have been
widely documented in computational fluid dynamics literature (Weatherill 1998),
limited efforts have been made to generate them using GIS feature-objects for
hydrologic modeling applications. This is in part due to the limited use of
unstructured meshes in hydrologic modeling, which have been restricted to TINs
only (Ivanov et al. 2004) and also because of the disconnect between data structures
of the GIS feature objects and geometric objects used in computational geometry
applications. Most of the unstructured meshing tools (e.g. Henry and Walters 1993,
Shewchuk 1997) use points or planar straight line graphs (PSLGs) to generate
Delaunay triangulations. This necessitates representation of existing GIS feature
objects such as points, lines, polygons, junctions and edges as either points or
PSLGs only. A PSLG is a set of vertices and segments that satisfies two constraints:
(1) PSLG must have two vertices that serve as endpoints; (2) segments in a PSLG are
permitted to intersect only at their endpoints. By reprocessing, we can essentially
reduce all the different feature objects to either a node or a line, making them
suitable for use in traditional unstructured meshing tools.
4.1 Reprocessing of GIS objects
Data structure of points and junctions can have a representation similar to nodes
when both are defined with an identifier, a coordinate location and a corresponding
attribute. Edges and polyline features in GIS have data structures similar to PSLGs
when defined by an identifier, number of segments in each compound object, and a
start and end node identifier for each segment. Edge features also carry additional
topology information. A polygon feature can be viewed as a collection of chained
polylines that form a closed boundary around individual areas having no gaps or
overlaps (Nyerges 1993). Figure 3 shows the data structure of each object type. A
GIS representation of a natural watershed is a complex multipolygon feature
composed of subshed boundaries (Figure 2), lakes, wetlands and river reaches. For
application to unstructured meshing tools for watershed modeling, multipolygon
features need to be further broken down into simplified polyline features. This is
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accomplished by disintegrating each polygon feature into polylines at junction
(intersection) points or at the entry node of a polygon. The process involves a
sequence of four steps: (1) identify polygons that share edges; (2) identify the
junction nodes of sharing polygons; (3) store polyline segments between the two
junction nodes or between a junction and an end/start node; (4) discard duplicate
polyline obtained from either one of the sharing polygons. Identifying the total
number of polygons as totPolygon, the pseudocode for transforming multipolygon
into polylines is shown below:
PolygonToPolyline( ):
For i50 to totPolygon {
For j50 to (i21)th Polygon {
If ((MBD of ith Polygon) > (MBD of jth Polygon)) ? NULL {
IdentifyJunctionPoints()
Disintegrate() Polygon to PolyLines at junction point(s) and at id 5 0
Delete() shared polyline from jth Polygon
}
}
}
Appendix:
If ((MBD of ith Polygon) > (MBD of jth Polygon)) ? NULL
R Intersection Area ?0
R Polygon i shares junction(s) with Polygon j
IdentifyJunctionPoints():
For k50 to numPts_1 in IntersectionArea of Polygon 1 {
For m50 to numPts_2 in IntersectionArea of Polygon 2 {
If (kth Pt15mth Pt2) { /* These points are shared by two intersecting/
partially overlapping polygons */
/* Following conditions identify a junction node from the shared nodes */
If ((((k21)th Pt15(m21)th Pt2) AND ((k + 1)th Pt1 ? (m + 1)th Pt2)) OR
(((k21)th Pt1 5 (m + 1)th Pt2) AND ((k + 1)th Pt1 ? (m21)th Pt2)) OR(((k + 1)th Pt1 5 (m + 1)th Pt2) AND ((k21)th Pt1 ? (m21)th Pt2)) OR
(((k + 1)th Pt1 5 (m21)th Pt2) AND ((k21)th Pt1 ? (m + 1)th Pt2))) {
kth Pt1 and mth Pt2 are Junction Points
Figure 3. Data structure of node, polyline and polygon feature objects. NumSeg5Numberof line segments in a polyline. Seg. ID5line segment ID. NumPolyL5number of polylines.PolyL. ID5polyline ID.
1574 M. Kumar et al.
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}
}
}
Legends:
numPts_*: Number of points of Polygon * in the Intersection Area
Pt*: Nodal points belonging to polygon *
MBD: Minimum Bounding Rectangle
A schematic application of the algorithm on a representative multipolygon feature
is shown in a series of steps in Figure 4. The algorithm has been incorporated in the
implementation of PIHMgis (http://www.pihm.psu.edu/pihmgis/).
With all GIS feature objects being reduced to a point or a PSLG, a domain
decomposition tool like TRIANGLE (Shewchuk 1997) can be used for
unstructured mesh decomposition. However, the raw polylines and the ones
obtained from reprocessing of GIS polygons generally have segment lengths that
are very small compared to the dynamical scales of interest in the hydrologic
model. The smaller segment lengths are an artifact of DEM-processing based
watershed delineation algorithms that are available in ArcHydro (Maidment
2002) and TauDEM tool (Tarboton and Ames 2001). Segments obtained from
DEM processing have their lengths determined by the resolution of the DEM used
for watershed delineation. Digitized watershed polygon boundaries may be
composed of segments with smaller lengths than are needed to efficiently represent
process dynamics. Smaller segment lengths translate to generation of unnecessa-
rily small triangles in the vicinity of polylines (Figure 5), thus also resulting in an
excessive number of mesh elements in the model domain. This places an
impractical time-step restriction on the model to maintain numerical accuracy and
stability [e.g. the Courant–Friedrichs–Lewy (CFL) condition in explicit time-
stepping methods, Courant et al. 1928]. In order to produce high-quality meshes
Figure 4. Intermediate steps in polygon to polyline simplification.
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while still maintaining the fidelity of the internal and external polyline boundaries,
reconditioning of polyline needs to be performed.
4.2 Polyline reconditioning
Reconditioning of polyline translates to a simplification of the line boundary by
removing small fluctuations or extraneous bends while preserving the essential
shape. The approximation error here is controlled by setting a maximum-distance-
from-vertex to the approximated polyline criteria. Simplification of polyline is
performed based on the Douglas and Peucker (DP) algorithm (1973). The first
approximation of the polyline is a line segment connecting the first and last vertices
of the polyline. Recursively, the maximum normal departure of each segment of the
approximation to the vertices in the original polyline segment is calculated. If the
distance from the farthest vertex to this approximate segment is larger than some
specified tolerance, then the vertex is added to the approximating polyline. The
algorithm terminates when all vertices in the original polyline are within a specified
tolerance for distance. Sequential execution of the algorithm on a representative
polyline is shown in Figure 6. Pseudocode of the algorithm is shown below:
SimplifyLine( ):
Read points in polyline from 0 to n: Pts[node_0, node_1,…………node_n]
Figure 5. The bottom left and right figures show unstructured mesh decomposition of LittleJuniata Watershed before and after reconditioning of polyline. The bottom two figures showthe zoomed-in fused image of triangulations at two different locations in the watershed. Notethe excessive high concentration of triangles formed in the ‘unconditioned’ case. Polylinereconditioning removes aliasing in the boundaries by removing nodes at unwanted locations.
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/* Recursive simplification. Starts with selection of end points of a polyline */
Simplify (Pts [], tolerance, 0, n)
Connect all MARKEDnode to get Simplified Line
Simplify (Pts [], tolerance, i, j) :
/* Find the vertex in original polyline that is farthest from approximate segment */
If (j + 1.k) {
MAXdistance 50
For index 5 i + 1 to j21 {
Find distance() of node_index from line segment (node_i « node_j)
If (distance() . MAXdistance) {
MAXdistance 5 distance()
MARKnode 5 index
}
}
/* Use the MARKEDnode (that is farthest from approximate segment) to
anchor another set of approximation */
If (MAXdistance . tolerance) {
MARKEDnode 5 MARKnode
Simplify(Pts[], tolerance, i, index)
Simplify(Pts[], tolerance, index, j)
}
}
/* Distance calculation between node_index and segment(node_i « node_j) */
/* node_index 5(x3, y3);node_i5(x1, y1);node_j 5 (x2,y2) */
distance():
m~x3{x1ð Þ x2{x1ð Þz y3{y1 y2{y1ð Þ½ �
node i{node jk k2
Figure 6. Intermediate steps in polyline simplification using Douglas–Peucker algorithm.
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Distance~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x3{ x1zm x2{x1ð Þð½ �2z y3{ y1zm y2{y1ð Þð½ �2q
Figure 5 shows how polyline reconditioning results in a reduction in the number of
triangles formed particularly near the boundaries. The algorithm has been
incorporated in the implementation of PIHMgis (http://www.pihm.psu.edu/
pihmgis/).
4.3 Delaunay triangulation based mesh generation
Using Very Important Points (VIPs), junctions, observation station nodes and the
boundary PSLGs as constraints, unstructured meshes that satisfy an ‘empty circle’
condition are generated. Such unstructured meshes are called Delaunay triangles.
We note that an ‘empty circle’ condition means that the circumcircles of the triangles
does not enclose the vertices of any other triangles in the mesh. Delaunay
triangulation also satisfies ‘max–min angle optimality’ condition (Lawson 1977).
This optimality criteria essentially ensures a high quality triangulation, i.e. triangles
created have a circumradius-to-shortest edge ratio (say, M) as small as possible
(Miller et al. 1995). This means that the generated triangles are balanced and not
skewed. VIPs that are used as constraints during triangulation are obtained by the
implementation of VIP algorithm of Chen and Guevara (1987) or Fowler and Little
(1979) on a DEM. External boundary polyline, which can be either concave or
convex, acts as a spatial limit-constraint in decomposition, beyond which no
triangles are generated. The tolerance criteria in polyline reconditioning and VIP
algorithms determine the degree of approximation of vector boundaries and raster
DEM respectively. Smaller tolerance criteria results in tighter approximations but it
also increases the number of unstructured meshes. Figure 7 shows the generation of
higher resolution meshes with a decrease in polyline tolerance criteria and increase
in number of VIP nodes. For all unstructured mesh decomposition resolutions,
triangulations outperform their structured mesh counterparts with the same average
resolution in terms of both efficiency and accuracy. Figure 8a shows the average
root-mean-square error (RMSE) in representation of DEM and its slope by a
structured grid decomposition of the watershed relative to an unstructured mesh.
RMSE is calculated by evaluating the difference in elevation/slopes for both
structured and unstructured grids w.r.t original high resolution DEM. Three
decomposition levels for unstructured meshes used in calculation of the error
statistics correspond to the decompositions shown in Figure 7. The average
structured grid resolution (Rsg) is calculated by
Rsg~
ffiffiffiffiffiffiffiffiffiffi
A
Numn
s
ð1Þ
where A is the area of the watershed and Numn is the number of nodes in the
unstructured mesh. By this simple measure, unstructured meshes outperform
structured meshes at all resolutions and for all criteria considered. Figure 8b shows
the average vector error per unit length of original polyline boundary in capturing
its tortuousities by structured and unstructured grids respectively. Vector error is
calculated by calculating the area of the enclosed polygon bounded by original
polyline and the corresponding boundary polyline representation used in structure
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or unstructured meshes. Again unstructured meshes show better performance for
the representation of boundary heterogeneities. Obviously with increasing resolu-
tion of either type of grids, there is a reduction in the error magnitude. We note that
a standard Delaunay triangulation is not sufficiently suited for hydrologic modeling
as it does not ensure generation of ‘quality’ triangles and also does not respect the
internal boundary constraints.
5. Model and data constraints on decomposition
Depending on the numerical formulation of the physical equations, local
topographic gradients and physiographic heterogeneity, generation of unstruc-
tured meshes can be customized to enhance representational accuracy and to
capture different time scales of process interaction. Customization criteria are
generally based on constraints posed by hydrologic model design and data
heterogeneity.
5.1 Model constraints
Many of the unstructured mesh based models like PIHM (Kumar 2009, Qu and
Duffy 2007) and RSM (Lal and Zee 2003) assume that the flux across the triangle
edges is always orthogonal. This assumption simplifies the numerical formulation of
process equations and also saves extra computation in defining the directional
Figure 7. The number of mesh elements increases with increasing number of VIPs insertedduring Delaunay triangulation and the decreasing tolerance magnitude used in polylinereconditioning of boundaries.
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components of the fluxes. The ‘orthogonal’ condition is ensured by considering a
circum-center (instead of centroid) as the representative location of triangular
elements. We note that the line joining the circumcenter of neighboring triangles will
always be perpendicular to the common side shared by the triangles. Triangulations
generated based on a circumcenter formulation also aids diagonal dominance
(Baker et al. 1988) and faster convergence of the numerical method (Vavasis 1993).
However, as shown in Figure 9, the circum-center of obtuse angled triangles can
sometimes fall outside the triangle. To avoid this condition requires a constraint on
the triangle shape that all the angles have to be acute. It is evident from Figure 10
that the upper bound on the ratio M (circumcenter-to-shortest triangle edge) ensures
that the lower bound on the smallest angle of the triangle is sin21(1/2M) and vice
versa (Shewchuck 1996). The lower bound on the smallest angle in turn bounds the
largest angle of the triangle also, i.e. if the smallest angle is h then the largest angle
cannot be larger than 18022h. This implies that by manipulating the value of M, we
can have Delaunay triangles with its largest angle bounded. Ruppert’s Delaunay
refinement algorithm (Ruppert 1995) employs a bound of M~ffiffiffi
2p
which means the
angles of the triangles range between 20.7 and 138.6u while Chew’s Delaunay
Figure 8. (a) Root mean square error in representation of DEM and slope at three levels ofmesh decomposition. Decomposition levels (a–c) are the same as those shown in Figure 7.Structured grids based decomposition leads to larger error in both DEM and sloperepresentation than unstructured grids. (b) Root mean square error in representation ofpolyline per unit length of polyline. At all decomposition levels, unstructured grids agree tothe boundaries better than structured grids. SrG5structured grid decomposition,UnSrG5unstructured grid decomposition.
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refinement algorithm (Chew 1993) employs a bound of M51 which means the
angles range between 30 and 120u.Another model constraint desirable for unstructured mesh generation is size. The
maximum allowable size of the triangles determines the total number of discretized
elements and hence the computational load and memory storage requirements for a
simulation. Furthermore, meshes are expected to have the ability to grade from
small to large elements over a relatively short distance. A larger value of M
translates to sharp gradation in triangle size. A Delaunay triangulation algorithm
proposed by Bern et al. (1994), Baker et al. (1988) and Hitschfeld and Rivara (2002)
generates non-obtuse triangles only, perfect for use in circumcenter based model
formulations. Nonetheless, these algorithms have limited flexibility in terms of
controlling the upper bound on the number of nodes in triangulation, the
implementation of VIPs and PSLGs as internal boundaries and also in terms of
spatial gradation in triangle size. The user must settle for a tradeoff between the
number of elements in the watershed and the number of triangles that violate the
non-obtuse criterion. Ruppert’s algorithm (Ruppert 1995) generates nicely graded
Figure 10. The upper bound on M (ratio of circumradius to the smallest triangular edge) iscontrolled inversely by an angle a that is subtended by the smallest side AB of the triangularelement on the opposite vertex.
Figure 9. Circumcenter O of DABC will lie inside its boundaries if and only if the triangle isacute angled, i.e. a,90.
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and optimal-size triangles while relaxing the non-obtuse triangle criteria. This
algorithm produces a mesh whose size (number of elements) is at most a constant
factor larger than the size of the smallest possible mesh that meets the same angle
bound (Shewchuk 1997). For the relatively small percentage of triangles that violate
the non-obtuse criterion, the centroid is assumed to be the representative of
triangular element.
5.2 Data constraints
Thematic spatial data that relate hydrographic and hydrogeologic parameters (e.g.
soil maps, stream cross-sections and wells/sinks) (Peuquet 1988) can be used as
constraints on the generation of unstructured meshes. These datasets can be
irregular or regular sample points, contours, polygons, grid cells and triangular nets.
All these types of datasets can be handled as one of the three kinds of constraining
layers (Bern and Eppstein 1992) in domain decomposition as discussed earlier.
5.2.1 Internal polygons/polylines. Reprocessed internal polygons and polylines can
serve as an internal boundary for triangulation which essentially means that they
can support triangulation on either side of boundary. The triangulation generated
while taking into account the polyline PSLGs as constraining boundaries is called a
constrained Delaunay triangulation (CDT). Internal boundaries pose an extra
constraint for Delaunay triangles as their interior cannot intersect a boundary
segment and their circumcircles should not encloses any vertex of the PSLG that is
visible from the interior of the triangle (Shewchuk 1996). Two points are said to be
visible to each other when no PSLG lies between them. CDT has been also used by
Vivoni et al. (2005) while using landscape indices as the PSLG. However,
constrained Delaunay leaves some of the triangular elements adjacent to the
PSLG to be non-delaunay. By the addition of extra Steiner points on the PSLGs,
such triangles are transformed to follow a conformed Delaunay Triangulation
property while still respecting the shape of the domain as well as the PSLG.
5.2.1.1 Typical internal polygons: thematic classes and hydrodynamic
descriptors. Examples of internal polygon boundaries are thematic classes like soil
types and land use/land cover types or hydrodynamic descriptors like subshed
boundaries and hypsometry.
The unique advantage of using thematic classes as constraints in decomposition is
that the resulting model grid contains a single class. This leads to non-introduction
of any additional data uncertainty arising from subgrid variability of themes within
a model grid. Figure 11 highlights this concept, where an unstructured mesh
generated using soil theme as a constraint leads to decomposition where each
triangle has a single soil type. For structured grid decomposition with the same
average resolution as the unstructured mesh, we observe that 41.63 % percent of the
grids have mixed themes. Generating grids that do not follow edges of thematic
classes (as shown in the case of structured grids) introduces uncertainty in
parameterization and its effect is widely documented in hydrologic modeling
literature (Beven 1995, Yu 2000).
Hydrodynamic descriptors can be thought of as topographic controls, such as
subshed boundaries and hypsometry that influence the movement of water through
the landscape (Winter 2001). Implementing hydrodynamic descriptors as constraints
in domain decomposition (Figure 5) has several advantages in modeling:
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(1) precise evaluation of the magnitude of groundwater flux exchanges across
the subshed boundaries. Since the subshed topographic boundaries are fixed(and so are the mesh edges that are anchored to these boundaries), seasonal
shifts in the ground water divide and flux with respect to the subshed
boundaries can be tracked
(2) the specified no-flux condition on surface water flow across the subshed
boundaries reduces computation load
(3) it is evident from Figure 5 that the triangular elements along the boundaries
are smaller than internal elements. The subshed boundaries also represent
relatively higher regions in the watershed where the hydrologic gradient can
be expected to be high. Large gradients in elevation directly affect changes in
temperature, wind, precipitation and vegetation. In order to capture these
changes, relatively smaller sized meshing is needed. The approach to
triangulation of subshed boundaries outlined above helps to achieve this
(4) This approach also provides a multiscale framework of modeling the basin
at watershed and subshed scales simultaneously.
Figure 11. Top left: unstructured mesh decomposition (UnSrG) is generated based onthematic class (soil type) boundary as constraint. Top right: structured mesh (SrG)decomposition has the same spatial resolution as the grid on left. The colored grid in thebackground of both the decompositions is a soil type map. The zoomed-in image shows thatSrG (in light gray) have multiple soil classes within them. UnSrG edges [in red or dark gray(in black and white)] overlap soil class edges, thus resulting in a ‘one soil class assignment’ toeach triangle.
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Another hydrodynamic descriptor is hypsometry. Shun and Duffy (1998)
observed that a simple three-region hypsometric classification of watershed area
into upland, lowland and middle elevation intervals could capture important
elevation changes in precipitation-temperature-runoff. The uplands generally act as
recharge zones, intermediate elevations as translational zones and lower regions as
discharge zones. Hypsometry, or area–altitude relationship, relates horizontal cross-
sectional area of a drainage basin to the relative elevation above base level (Strahler
1952). Strahler (1958) interpreted regions with low hypsometric values as eroded
landscapes and high values as young landscapes with low erosion. This means that
polygons corresponding to the three hypsometric divisions can be assigned unique
attributes according to their characteristic time scales of groundwater flow or
geomorphic evolution. Separating boundaries can be treated as contours. Figure 12
shows the hypsometric curve for Little Juniata Watershed. The three hypsometric
divisions are obtained using the Jenks’s optimal classification strategy (Coulson
1987, Jenks 1977). With this method, intra-class variance of elevation values is
minimized and the differences between classes are maximized resulting in better
representation of groupings and patterns inherent in the dataset. As is evident from
Figure 12, regions in the basin that are relatively flat have larger triangular elements
Figure 12. Left figure shows the elevation hypsometric curve of Little Juniata Watershed.Delaunay Triangulation of Little Juniata Watershed while using hypsometric division as aconstraint. Expectedly in regions of higher topographic extremes/gradient, concentration ofmeshes is higher. Note the formation of smaller triangles besides the streams. Similardivisional constraints can be used for vegetation and climate regimes.
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whereas the mountainous regions are discretized into small elements. Traditional
slope preserving meshes that retain high nodal density in regions of high terrain
variability (Lee 1991) are generated when the constraints are derived from elevation.
5.2.1.2 Typical internal PSLGs: river network. An important example of
constraining PSLGs used in domain decomposition is a river network. A higher
concentration of triangles generated along the river network (Figure 2) reflects the
relatively faster hydrodynamics of riparian regions. Accurate assessment of flooded
areas that get inundated with slight increase in river head can be performed when
smaller triangles are generated in the vicinity of the river network. This facilitates
flood-plain and flood inundation zone mapping.
We note that with the increasing number of internal boundary constraints, the
number of mesh elements increases. This implies that a tradeoff exists between
accuracy of representation of watershed properties (which is gained through the use
of internal boundary constraints) and computational load. While the choice of any
of these constraints is optional, any decision to insert an internal boundary must
take into account the tradeoff between accuracy and computational load. Assuming
that there is no uncertainty associated with the location of internal boundary itself,
quantification of accuracy gained after the use of an internal boundary can be done
at (1) ‘pre-modeling’ stage, where accuracy in representation of a watershed
property on decomposed domain is assessed, and (2) ‘post-modeling’ stage, where
accuracy in representation of modeled physical states such as evapotranspiration,
spatial distribution of snow, etc., are assessed. For particular boundary types (like
soil and vegetation), the uncertainty associated with their position and degree of
transition can be significant. In such situations, representational accuracy
calculations must be weighted by the inherent positional uncertainty. The error
posed by such uncertainty can be limited to a certain extent by using a buffer region
(and the boundary) on the either side of the boundary as constraints. Buffer
constraint will lead to generation of smaller mesh elements on either side of the
boundary until some specified width quantified by ‘position/transition-uncertainty’.
In such cases, mixed transition properties can be specified to the cells inside the
buffer while using ‘hard’ categorical properties outside of it.
5.2.2 Holes. The data structure of a hole is exactly that of a polygon. But there is a
constraint on the operation. Holes are regions inside the basin boundary or other
polygons that act as an external boundary and so are not triangulated. One example
of a hole is a lake. Figure 13 shows the Great Salt Lake and Utah Lake within Great
Salt Lake basin regions. Assuming that the height of water in the lake is the same
everywhere within its boundary, the lake can be considered as a single control
volume entity for modeling. This of course saves computational load where the
assumption is valid. For a model scenario of increasing lake levels from a minimum
pool, the adjacent elements will be submerged. However, because of the relatively
small triangles adjacent to the boundary, a more accurate depiction of temporal
changes in spatial extent of lake boundary and inundated areas can be performed.
5.2.3 Points. Constraining points along with the Steiner points act as vertices for
triangulation. A Steiner point is a node that is inserted in a line to divide it into
smaller segments. It is not a part of the original set of constraining points and is
generated only during triangulation. A typical constraining point can be the stage
observation stations at hydraulic control structures like weirs, gates, pumps or
ground water measuring stations. Modeled results of state variables at these
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constraining points can be compared directly to the observed values. This reduces
any uncertainty in comparison of model results to observation as it happens in most
cases where observation stations are not exactly the modeled locations. Figure 14
shows mesh decomposition for two cases, one with observation stations asconstraint and the other without it. We note that the observation station on the
river, as well as in the watershed, is left to dangle somewhere in the middle of the
discretized element in the unconstrained decomposition case. For comparison with
stage or groundwater level obtained from the model in such cases will need
interpolation of modeled results to observation locations. On the other hand, in a
constrained decomposition case, observation stations can also act as a node for
triangulation resulting in simulation of state variables exactly at the modeled
locations. To put it simply, modeled location and observation locations are the samein the constrained decomposition case. In the future, constrained domain
decomposition based on the sensor-networks locations should facilitate direct
assimilation of observed data into the model (Reed et al. 2006).
6. Advancements in hydrologic modeling
Using the constrained decompositions discussed earlier, integrated modeling of
hydrologic processes in Little Juniata Watershed (Figure 2) is performed. For
detailed model related information, readers are referred to Kumar (2009). The
model generates a large amount of spatio-temporal data of each hydrologic state
such as interception storage, snow depth, overland flow, ground water depth, soil
moisture, river flow and evapotranspiration. Figure 15 shows a representative
spatial distribution of modeled evapo-transpiration at two snapshots in time. Wealso show the spatial distribution of average river streamflow, percentage of time
different sections of the river is gaining and finally the streamflow time series at four
Figure 13. Domain decomposition of Great Salt Lake Basin. Note that no triangles arecreated inside the lake.
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separate locations in the watershed. These are few of the spatio-temporal predictions
obtained from model simulation. The model results shown in Figure 15 are from a
static decomposition of the watershed. A higher resolution modeling with least
additional computational burden can be performed by (1) selective ‘zooming’ in an
area of interest while leaving discretization in the rest of the watershed to be coarse
and (2) adaptively changing the grid resolutions in different parts of the watershed
depending on the spatial rate of change of a physical state in a localized region.
Figure 14. (a) Zoom-in of mesh decomposition with and without using groundwaterobservation stations as constraints for two locations inside the watershed. (b) Meshdecomposition with (right side, green) and without (left side, gray) using observation stationsas constraints. (c) Zoom-in of mesh decomposition with and without using stage observationstations as constraints for two locations on Little Juniata River. We note that in constraineddecomposition, triangulations are generated such that the observation stations lie directly onthe mesh nodes.
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Such advanced modeling is facilitated by the flexibility in generation of
multiresolution nested and adaptive (refinement/derefinement) unstructured meshes
while using different constraints.
6.1 Nested triangulation
Nested models are already common in other fields of science (e.g. fluid dynamics:
Carey et al. 2003; geology: Gautier et al. 1999; climatology: Loaiciga et al. 1996). In
hydrologic modeling, development of nested models is still in its infancy stage
(Grayson and Bloschl 2001). The basic principle behind this strategy is triangulation
Figure 15. (a) shows the average evapo-transpiration (ET) time series for the Little JuniataWatershed for 2 year period. Snapshots for spatial distribution of ET during the maximumand minimum extremes are shown in (b) and (c) respectively. Since we have used the samecolor range to represent both extremes, ET appears to be uniform everywhere (though that isnot the case) during winter as the values are quite small. (d) and (e) shows the streamflowhydrograph at four locations in the stream network. (f) shows the percentage of time eachstream segment gains water from the aquifer. We note that all the results shown above are fora simulation period of 2 years ranging from November 1983 to October 1985.
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of the domain of interest into large elements combined with locally refined or nestedgrids where higher resolution is necessary. The main advantage of the approach is
seamless assimilation of spatially varied forcings, parameters or the constitutive
relationships in different regions of the basin. This nested-grid configuration makes
it possible to combine realistic large-scale simulations with mesoscale forecasts for
selected regions. Figure 16 shows that triangles generated in a subshed around the
main stem of Little Juniata River are much smaller than the rest of the basin. This
local high-resolution representation of parameters and processes minimizes
computation. Nested triangulations will have applications for:
(1) integrated hydrologic studies in larger basins which have high resolution
data-support in one of its subwatersheds. The subwatershed in this case can
be decomposed at relatively higher resolution than the rest of the basin totake maximum advantage of the high resolution observed data while still
maintaining all boundary conditions and conservation rules
(2) studies which focus on understanding scaling issues and comparison of
scaling effects across the basin
(3) the implementation of new physical processes in watershed and river basin
studies which are relatively more computationally intensive. One example
might be physics-based snows melt modeling. Modeling snow-melt is quite
critical in mountainous basins as it directly affects flooding, contaminant
Figure 16. Nested mesh decomposition of Little Juniata Watershed while using subsheds asinternal boundary. For computational efficiency, a localized region of the basin (around mainstem of Little Juniata River, shaded in the figure) can be discretized to higher spatialresolution elements while leaving the rest of the basin at coarser resolution. Under a singleframework, mesoscale to microscale modeling can be performed.
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transport, water supply recharge and erosion (Walter et al. 2005). Snow-
melt modeling generally falls in two categories: temperature-index models
that assume an empirical relationship between air temperatures and melt
rates and energy-balance models that quantify the melt amount by solving
energy balance equations. The most common justification for temperature
index snowmelt models is the need for a reduced number of input variables,
and also the model and computational simplicity. Several researchers
(Ambroise et al. 1996, Fontaine et al. 2002) have derived acceptable results
from temperature index models. Nevertheless, because of the incorporation
of energy interaction between topography, wind and radiation, an energy-
balance model is better placed in simulation of spatial heterogeneity in snow
accumulation and redistribution (Winstral and Marks 2002). Since energy-
based models like SNOBAL (Marks et al. 1998) run at shorter time
intervals, need a high resolution DEM and forcing data as their support
and solve a large number of state variables, they can only be run over a
small watershed with present computational constraints or run in an offline
weakly coupled mode. So an energy based modeling can be performed in the
high resolution nested watershed while temperature-index modeling
approach can be used for rest of the watershed
(4) watersheds susceptible to large and frequent floods in limited areas but
where the runoff is generated in upland non-flooding areas. Higher
resolution discretization in a watershed will lead to accurate mapping of
areas which will get flooded with given increase in stream-stage level
(Shamsi 2002)
(5) understanding the relationship of groundwater flow dynamics within a
subshed to the rest of the basin. Subshed topographic boundaries are simply
a surface descriptor and as noted by Brachet (2005), the groundwater flow
divide can change seasonally affecting the water flux in and out of the
watershed. With a nested modeling approach, movement of ground water
divide can be mapped with acceptable precision.
Clearly, the nested triangulation strategy has implications for studying the scaling
behavior and scale transitions with a river basin.
6.2 Adaptive refinement/derefinement of triangulation
The triangulation and domain representation discussed above is spatially adaptive in
the sense that the resolution of the elements are not uniform. The resolution and
distribution of triangles depends on the constraining layers, the tolerance in VIP
generation and the polyline reconditioning algorithms, and the boundary hetero-
geneities. Clearly, widely varying spatial and temporal scales, in addition to the
nonlinearity of the dynamical system, can raise interesting and challenging modeling
problems. In many applications, because of the particular dynamics of the problem,
meshes may need to be further (de)refined locally after initial computation. An
example is when the computed solution is rapidly varying in time within small areas of
the domain while the solution is relatively slow in other parts. Solving such a problem
more efficiently and accurately requires a solution-adaptive triangulation strategy.
This is a necessity for resolving coupled physics with different time scales which is
encountered in hydrologic models. Solution adaptation can save several orders of
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magnitude in computational load by avoiding under-resolving high-gradient regions
in the problem, or conversely, over resolving low gradient regions at the expense of
more critical regions. While this strategy can prove problematic within finite
difference and finite element analyses, finite volume methods, particularly those based
on a triangular mesh system, lend themselves quite naturally to automatic adaptive
refinement/derefinement procedures (Sleigh et al. 1995).
Dynamically adaptive grid approaches have long been used in astrophysical,
aeronautical and other areas of computational fluid dynamics problems (Berger and
Oliger 1984, Berger and Colella 1989). Zhao et al. (1994) used a Riemann solver
approach to calculate fluxes on unstructured, mixed quadrilateral-triangular grid
system in river and flood plain. Sleigh et al. (1998) used adaptive refinement/
derefinement for computational efficiency while predicting flow in river and
estuaries using unstructured finite volume algorithm. For a finite volume integrated
hydrologic model like PIHM, adaptive refinement can be carried out in the regions
where intermittent dynamics occurs. An example is an ephemeral channel. Since the
channel is dry for most of the year, there is no need to have an a priori high-
resolution grid. Depending on the status of the channel network, the dynamics will
trigger adaptive refinement/derefinement. Other situations include Hortonian
runoff, convective precipitation in a part of basin and sudden increase in river
Figure 17. Flow chart depicting the dynamically adaptive refinement/derefinement algo-rithm for hydrologic modeling. Depending on the hydrodynamics, a particular region can berefined to finer or coarse triangular elements in order to capture the hydrologic processaccurately.
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stage due to dam break or flash floods. In each case adaptive refinement/
derefinement of existing triangulations will save on the computation load. Figure 17
shows the flow diagram depicting the refinement/derefinement procedure.
Refinement is performed by inserting carefully placed vertices until the triangular
element meets constraints on element quality and size which is decided a priori.
Inserting a vertex to improve elements that do not satisfy the refinement criterion in
one part of a mesh will not unnecessarily perturb a distant part of the mesh that has
no ‘bad’ elements. That is, insertion of a vertex is a local operation, and hence is
inexpensive except in unusual cases. The refinement/derefinement criterion is
generally based on metrics that quantify a sudden change in magnitude of a state
variable over a localized area or time. Figure 18 shows refinement of two marked
triangles in different parts of domain. It is obvious from the figure that the original
triangulation is affected locally only.
7. Conclusions
A strategy for unstructured mesh decomposition is proposed which captures the
phenomenological and hydrologic complexity of the watershed while minimizing the
equations to be solved. The framework provides a tight coupling between geodata
and the processes that are modeled on it. The strategy seamlessly incorporates
computational geometry based algorithms to process GIS feature objects for
discretization for model domain. The framework incorporates the constraints posed
by hydrologic process dynamics, numerical solver, data heterogeneity and computa-
tional load. It outperforms structured grids based representations in terms of accurate
representation of raster and vector layers. Polygon reprocessing and polyline
reconditioning algorithms facilitate the use of available GIS feature objects in
domain decomposition. Flexibility of the framework in terms of model implementa-
tion, model development, data and process constraints, and elevation-derived-VIPs,
Figure 18. Coarse-scale unstructured mesh decomposition of Little Juniata Watershed (left).At any time t during simulation, two triangles (light gray) are marked as Bad Elementsdepending on spatial gradient estimate of a state variable and are identified for refinement.Decomposition on the right shows insertion of a node inside the marked elements and theresulting perturbed region. Note the formation of new triangles and the triangulated area thatgets perturbed is very small relative to the whole watershed. Triangles in the unperturbedregion remain the same.
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provides added advantages when compared to traditional TINs. Rapid prototyping ofmeshes which better reflect the constraints of the problem under consideration can be
obtained. The problem constraints that are addressed include: the computational
burden, the need for reduction in uncertainty of state variables, the accurate
specification of boundary conditions, the application of multiscale or nested models
and the need for dynamically adaptive refinement/derefinement.
In summary, the ‘support-based’ domain decomposition and unstructured grid
framework provide a close linkage between geo-scientific data and complex
numerical models. The strategy extends the GIS based algorithms to be used indistributed numerical modeling setting. The framework is generic and can be
implemented for linking other numerical process models of mass, momentum and
energy to their respective geodatabases.
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